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OCR for page 540
On Submerged Stagnation Points and Bow Vortices
Generation
L. R~eja (Indian institute of Technology, Khara Dur, India)
AflSTltACT
Th mechmi m of the genemti m of bow
vertices in two dbm nsi ms, m hbomtory scale, is
expkmed on th basis of th :xistence of c
subm rged tagrution pomt below fhe fiee smtcce
Th t~eam which originctes from fhe submerged
tagrution poi t m fhe fie smtcce di ecti m reverses
md neut~ali es th mom nt m of mcommg flow
~esultmg m c flee su face separction pomt Th
miticl location of fhe submerged stagmtion pomt md
subsequently fhe flee smiace separction pomt is
cclcubted for fhe case of c s misubmerged horizontal
cimukr cylmd r, md fhe ktter is compmed wifh th
experime tal results The cgrement is ~ecsonably
good Th generction of bow vo tices is ddscussed as
c balance betweff~ mertial md gravitational effects
Th loss of pressure et fhe b w md th consequent
d cg due to b w vertices phffmm mr is calcuhted
md is fommd to cgre w 11 with th value fommd by
exper ime t A medho do logy for th de sign of
efhcient b w contour shap in two dimff~siom whe~e
fhe subm rged stagnati m point is used es c conhol
hmdle, is p~esented The fheory is clso mplied to
nm~egmbr shcpes Ike vetical step md bulbous
b w Th results are compared with fhose obtamed
by flow computati m md found to be m ~easonably
close cg~e ment Firully, c conject re is cd cnced to
expkm th genemti m of bow vo tices m the
dimff~siom, i e fhe neckk e vo t :x arommd c ship's
load waterlme, on the same basis
INTRODUCTION
Th vo ticcl moti m observed checd of c
particlly submerged object tow d m c hyd odynamic
tmk is k ow as bow vertices The bow vo tices
~egion is separcted fi m fhe mcm potential flow by c
shc,m boundary termed es fhe flee su face separcti m
poi t FSSP), when fhe flow is two dimem iorul Th
hyd odynamic flmmes The understmdmg of fhis
ph n menon is of di~ect comeq ence to bow wave
b~eaki g, which is re pomible for c substmticl
c mpone t of c ship's resirtfmce Bcbc 1969) The
ph n menon d picts itself in the form of white water
et th bow contmumg cll arommd the ship's load
waterlme, md is also called es neckk e vo t :x
Seve~al mfhors hcve isuclised bow vortices
checd of two dimensiorul es w 11 es thee
dimff~sionai shap s, eg E k t md Sharma(1970),
Smuki (197~9, Honji, (1976, Shchshalkm (1951),
b'~yo md Tckekumc (1951), b'~yo, Takekuma,
Eggers, Sharmc (1952) md Mori (1954) But, h re
w shell be primarily concemed wifh the experiments
of b'~yo, Tckekumc, Eggers md Sharmc (I 952) on c
hori ontal semi subm rged cimukr cylinder
c mducted et Imtit t f r Schiflb m, Hcmburg it may
be relev mt to m nti m f tt fhe mfhor hcs iewed the
videotape of fhese e perim nts b fhese
e perim nts, c cacukr cylmder was tow d m c
hori ontal semisubmerged condbti m (two
dimff~sionai flow) md fhe bow vo tices w ~e
visualised usmg c watercolour dye Th FSSP was
mecsmed for diffe~e t vulues of d cit Froude mmmber
F~) Th primaryobjectiveofthispcp ristoe pkm
fhese ~esults qualitatively md q mtitatively But
befcre fnat, w shell p~esent c brief review of th
cttempts mcd so far in this du ectioa
Dcgen md Tulm (1972) solved th g~avity
fl w pc t c blmmt body by usi g two pert rbati m
e pensions A mcll Froud mmmber solution was
obtcined fcr the flow mmder the mmbrokff~ fiee smtcce
upto second crder, while c high Froud m mber
solution was obtamed based upon fhe mod I of c jet
detachmg fi m th bow md not retmomg to fhe fl w
The heakmg of fhe wave was cssigned to Tcylor
mstability d e to fhe tepff~mg of fhe sheamlmes A
OCR for page 541
critical F~ was obtcined to characterise th omet of
wa~ b~eaki g Th cssocicted d cg due to b~eaki g
of waves was also calcubted md was found to be
twice fhe value estimated by Bcbc (1969)
experime tally
Mori (1984) assigned fhe white water
gff~erction phenomffmn to she fl w imtability md
subsequent b~eakmg of fhe bow flow, md tudied th
same themeticclly md experime tally m g~ecter
detail The fiee smtcce cmvatme was concluded to
be one of th somces of sh ar fl w benecth th fie
surtcce Stability am~lysis, vcrticity shetchi g
fheory, md fiee su face boundary kyer fheob w ~e
mvolved to expkm fhe exp rimental results, eg
velocities, Rey old shesses md b w wave heights
Pctel, mdweber md Tmg (1984) cttempted to
expkm fhe b w vo tices genemtion on th basis of
fhe :xistence of c fie smtcce bommdary kyer, which
is c kyer of concenbated vorticity occunmg due to
fhe curvat re of fhe fie smiace m fhe fl w of c ~ecl
fluid Th cuthors sp cuhted f tt fhe fiee surtcce
would move slower f m th kyer benecth it md fhis
velocity defect would lecd to c FSSP checd of th
body md subseq e tly fhe bow vo tices Fu ther, by
assmmmg fnat et th fiee smiace th smtcce tensi m is
baLmced by ncrmal viscous shess fcrce, m
exp~essi m for FSSF locati m was obtcined by d6'ect
mtegmti m of fhe boundary c mdition using th
contim ity quctioa The locati m of fhe FSSF was
obtcined m terms of fhe slop of fhe fie smtcce
Th ~esults so obtamed were cpplied to c cimukr
cylmd r md compmed wifh th exp rim ntal values
of b'~yo et cl (1982) The fheoretical vulues show d
m inc~ecsi g t~end, ie fhe FSSF will move cway
fi m th body with mcrese in d cft Froud m mber
while exp rim ntal results pomted to fhe ~everse
H wever, fhe e perim nts of Grosenbmgh md
Yemmg(1985,1989) ~eported good cg~eem nt wifh
fheir separcti m criterion The idec of c fiee smtcce
boundary kyer lecd6 g to c FSSF was fmfher
examined by Rahqc (1995) The fie surtcce
boundary Icyer velocity md vo ticity proflles w ~e
computed et various stati ms up t~eam it was
observed thct fhe fie smtcce moved slower f m th
flow beneadh it but fhis velocity defect was not large
ffmugh to ~esult m c FSSF ahecd of th body Yeu g
md Ammthakrishnfm (1992), in their computaticrur
tudy of fhe probl m (to be dscussed hter), also
conclud d fnat th fiee smtcce vo ticity is not mtense
ffmugh to lecd to bow vo tices The bommdary kyer
vcrticity proflles computed by Rahejc (1995 poi ted
towards mstability of th boundary Icyer flow owl g
to fheu nommonot mic rutme
VmdeeBroeck & Tuck (1977), Vmdn
Broeck, Schwart md Tuck (1978) contmued
mvestigati ms mto the cnalytical solution of th bow
fl w problem by usmg c series e pension m Froud
m mber, but conclud d f tt it was not possible to
obtcin c co tmuous bow wave proflle becmse of
nm mmiqueness of th soluti m However, th bow
shape was reshicted to bows wifh c ve ticcl or
mclmed flct faced They pecubted thct fhe possible
fomm of soluti m for fhese shmes is thct of m
ovetmomg jet Tuck md VmdeeBroeck (1984)
show d fnat c co tmuous splashless bow flow was
possible fcr some diffe~e t bow shap s such as
bulbous bow it was cssum d cll though that th
tag,nction poi t lies et fhe mtersection of th body
md fhe fiee su face
The difhculty in fmdmg c closed for
solution gave m impetus to computatiorul st dies
Miyata et al (1985) mplied c version of th M C
method to ccpt ~e nmlinear wave breaking et th
b w The nonlmear waves b~eaki g et th bow due
to tepness w re temmed as fiee smtcce shock wave
FSSW) The exact n mlinear fiee smtcce condti m
was used et fhe flee bommdary md fhe no slip
c mditi m on fhe body A computational st dy, which
gives mme msight mto fhe bow flow, is fnat of
Grosenb mgh md Yemmg (I 989) These mfhors have
used c boundary mteg~al medhod to compute fhe two
dimff~sional flee su face flow pest c semi i fmite
body m th time domcia The fie smtcce
c mputation is done accordmg to th m fhod give by
Lo guet Higgms md Cokelet (1976 The criticcl
Froud mmmber F~ fcr fhe omet of wa~ b~eaki g is
found for bow shcpes vertical step, faued body md
bulbous b w It is fommd th re fnat c bulb m the b w
shape delcys the omet of wave b~eakmg In fhis
tudy, th flee surtcce fl w is developed fiom the
tecdy tate double body fl w th s avoidmg th
impulsive tart of the body which may result m
unco trolled bow wa~ elevatioa This is also
supported by th fmdmg of Dcgen & Tulm (1972)
where th low st order asymptotic expmsi m of th
fie smtcce flow pest s mi mflmte body is found to
be c double body flow wifh fhe fie smtcce repk cd
by c rigid pkte Besid s, it seems quite logical to
c msider c stecdy state double body flowas fhe miticl
c mditi m; fhe suddenly r mo i g fhe upper hclf of
fhe flow cllows th flow to develop to c fiee smtcce
fl w The mfhors also discuss th occm~ence of
submerged tagrution pomt(SSF) on th body, which
behaves differently for fhe b~eaki g regime md non
b~eaki g regime of fhe flow For th fcrm r case, the
SSF r mcms bel w fhe flee smtcce while th bow
wa~ overt ms, but fcr th ktter, fhe SSF is miticlly
bel w fhe fiee su face md rises to fhe flee smtcce as
OCR for page 542
fhe bow elevation ~educes md settles et th
tagrution height of O SF~2 We shell discuss fhese
~esults ktter in moth r section of th work
Yeung md A mbnaloish m (1992)
mvestigated fhe fl w of c ~ecl fluid past c two
dimff~sional bow, one of the cim bemg to examine
fhe possibility of b w vo tices observed et kborctob
sccle The mfhors concluded thct th occmrence of
b w vertices in th kborctcry scale is due to th
p~esence of surfactmts which accmmubte near th
body md pro id rigid bommdary I kc behcviom to
fhe fiee surtcce ledmg to boundary ky r separcti m
md subsequently vertices The N. 5 equctiom
coupled wifh th su factmt conce tmti m eqrcti m
w ~e computed by c varictiorul flactiorul tep
method md usmg fhe no slip condition et th fie
surtcce, fhe occm~ence of vo tices was show Th
mfhors clso mvestigated fhe case of fie slip md th
exact nonlmear flee smtcce boundary condition md
arrived et fhe conclusion thct vorticity genemted due
to fiee sutcce cmvatme is not i tff~se enough to lecd
to separction As c rece t development, Dong, b'~t
&Humg (1997) hcve usedPIV to visualise th bow
flow md mesured th flow velocities near fhe bow
wa~, upshem md dow t~eam The laser sh et is
visualised m differe t orientati ms checd of fhe bow
md et different stations dow sheam of fhe bow
Smmmarismg th above ~e i w, one may
menti m fnat fhe ideas of Tcylor in tability, fie
surtcce bommdary kyer md su fact mt conce tmti m
have ben emmmed but th mechmi m of bow
vertices generction md fhe occunence of fhe FSSP
are still not w 11 understood md to th be t of om
k owledge th results of fhe e perime ts by b'~yo et
al (1952) have not yet ben e pkmed A two
dimff~sional tudy is c msidered clmo t c necessary
tep m fhe developme t of c th °b for fhe f e
dimff~sional case, es it provides c valuable gam m
msight et fhe expff~se of rehtively simple
computatioa The~efme, it is desuable to conce tmte
on fmdmg c theob fcr e phim g th e perim ntal
~esults of b'~yo et cl (1952) befme discussmg th
f ee dimemiorul generction of white water,
neckdace vo t:x or b w wave b~eaki g in m oce m
goi g ship b fhe p~esent work, w propose c th °b
to expkm fhe bow vo bees genemtmn m th
kborctcry scale m two dimff~sional fl w on fhe basis
of th occunence of c subm rged stagnation pomt
(SSP) This is cnalyticclly found for fhe flow pest c
semisubmerged hori ontal cacular cylinder
Subsequently, th cppro imate vulues of th PSSP
for different d cit based Proud mmmbers me
cclcubted md compared wifh fhe e perim ntal
values of b'~yo et cl (1952) Pu th r, th d cg due to
loss of pressme et th bow is cclcubted md
c mpared wifh th value estimated by Bcbc (1969)
e perim ntally Besid s c cacukr cylmd r, th
fhecry is clso cpplied to c vertical tep md c bulbous
b w shme Th design of m effcient bow contom m
two dimff~siom is discussed Pinally, c conject ~e is
cd cnced fcr e pkmmg fhe f ee dimemiorul bow
vertices generctioa The subseqre t sections of fhis
pmer develop th idec md fhe relevmt expressiom
m c tep by tep manner ff~dmg with conclusi ms md
f t ~e scope of work
SSP THEORY AND BOW VORTICES
GENERATION
The mech ml m of bow vertices genemti m
md fhe occmrff~ce of th fie su face separcti m
poi t c m be exphined und r fhe fiamework of th
proposed SSP th °b as follows A tagrution pomt
m fhe two dimff~sional flow pc t c f lly submerged
object is clways fhe mtersection of th di id6 g
t~eamlme md the body The sheam divides itself
mto two parts et this pomt Th pressme on bodh fhe
sides of fhe tag,nction poi t d meases The pomt is c
maximum of fhe p~essure dish ibution b fhe case of
fhe flow pest c partially subm rged object, c
tagnation pomt may :xist below th fie su face md
may be rightly called c submerged stagnati m pomt
(SSP) es show m Pig I It may be conject ~ed fnat
m case m SSP does :xi t it should similarly be th
mtersection of fhe di id6 g sheamlme md fhe body
Accordmgly, the sheam should d6 id itself mto two
parts et fhis poi t md ff th fl w is two dimff~sional,
one part will flow below fhe object md fhe of her
should move upward m fhe d6'ection of fhe fie
surtcce This ktter part of th sheam, wi g to th
obvious limitation in mo i g upward due to g~avity,
~everses md neabalises fhe velocity of fhe incomi g
fl w ~esultmg m m PSSP where fhe two velocities
c me m baknce Th reversmg flow tmps c regi m
offhefluid(Fig l)extff~dmghori o tallyfrombody
to th PSSP md verticclly fi m the SSP to fhe fie
surtcce P, duly bommded by th d6 id6 g sheamline
md the fie su face such thct fhe fluid is mo i g
clo g fhe boundary of fhis regi m, i e fiom PSSP to
SSP clo g th d6 id6 g sheamline, from SSP to P
clo g fhe body md fmclly toward PSSP clong the
fie smiace The cucubtion of fhe fluid clong th
boundary of fhis regi m sets th entue mass of th
h mped fluid mto cyclic motion givmg rise to fhe flrst
vertex wifh positive i e mti clockwise vorticity et
fhe cenhe which may be due to only i terrul sh ar
The vo tex grows in si e md fcr stability reasom, c
see md vertex starts from fhe body sid with rotati m
m fhe opposite duecti m Thus fhe fluid k eps m
OCR for page 543
yl
-: ~t-
ssP ~'
Fig I Scbm rged shgnstioc poict (ssP) sad the sssocisted bow vordces gece~tioc ic the dow psst s hortrochl
semisubme ged circcEr cylinder
entermg m fhis regi m fi m fhe body side due to th
upward mo ing tt am, md fhe vo tices me
contimtously produced fi om body sid The boundary
FSSP fhus keeps on shiftmg slowly tmmmd upsheam
as th regi m is filled Fu th r, mwi g to this fluid
fllli g, th bow wa~ height contmuously mcreases
md upsets fhe inerticl g c itatiorui baknce which
t suits m e pelli g fhe flrst vo t :x out of fhe regi m
to jom th main flmw with fhe FSSF mm i g
backward md th bow wa~ height comi g dmvrt
simmltaneously, md fhe process of fllli g th regi m
t starts The oscillation m bow wave height md th
cort pondi g back md fo th moveme t of fhe FSSF
have ben obsetved e perime tclly by Grosenb mgh
md Yeu g (1955,1989) md clso by fhe present
mfhor m m op n~u cu thti g water charmel
(unpublished) Th s, fhe fluid keps on enteri g fhis
t gion from th upward mm mg sheam md th
vmtices are contmu tsly prodt cd from body sid
md expelled fi m fhe of her sid to jom th mcm
tt tm This, in c mttshell explaim th mech ml m of
bmw vmtices gff~erction md fhe occunff~ce of th
FSSF The sit ation described above cone p mds to
fhe case whert fhe fiee t face is not brokert in case
fhe ft e t face heks, fhe upward mm mg sheam
fi m fhe SSF fomms c bmw jet, which enhaim air md
dismteg ates to fmm bubbles, reversmg i to th mcm
flow cgain t sultmg m th FSSF Vmdet Broeck
md Tuck (1977) had specubted c simibr scenario as
one of th possibilities while cttemptmg to flnd m
am~lyticcl soluti m for verticcl or inclit d flat faced
bmw shcpes
FtESULTS AND DISCUSSION
SSP -Evisteue cud Locetiou
To confum the e i tff~ce of m SSF md
tbs qtt ntly fmd its position should t quit, m
genemi, solvi g fhe glavity flmw problem, which of
c trse, is very difft uit But it is possible to k ow
cbout fhe SSF md evert flnd fhe mitial location of th
SSF by amrlysi g the mitial c mdition of fhe probi m
Com ider fhe two dimertsimuri flow pc t c horizontci
semi tbm rged cit ukr cylinder it is cssum d ft~t
fhe fiee t face flow is obtamed by mitistly he i g
potential double body flow md removmg th upper
flmw suddertly at m imt mt (t = 0+) which may be
ttkert as the omet of gm ity flow (Grosenbmgh md
Yemmg 19S9) Hff~ce the pres tre md fhe velocity
fleid pt vaiimg imticily, i e et t = 0, me th s k mvrt
md it is possible to se wh fher m SSF e i ts md
cal thte fhe iocati m of th SSF which wiii th n be
fhe pomt wh re fhe total pt s tre wiii have its
maximum on the body it may be me tioned ft dt th
fie smtcce flow at t = 0+ is d tbie body flow
solution of fhe Lmk e quation wifh fhe pt s tt
t pkt d by total pt s tt, i e mciudi g gravity term
Referring to fhe axis sy t m es show t m Fig I, for
fhe tt am flowmg m fhe positive x dit ction, the
mitici pres tre di tribution, i e at tim t = 0+, at my
poi t (x, y) in fhe flmw wlli be given by totci pt s tt
~s
P=Pat +OSp(U~ U2) psy
(1)
OCR for page 544
wh re U is fhe velocity of the flow, md U~ is th
velocity far upshem md P~h, is ctmo pheric
p~essme
Fcr th pomts on fhe cylmd r, w k ow fi m th
~esults of pote ticl double body flow fnat
U=2U~ smO'
(2)
wh re O' is the polar coordincte of fhe pomt
mesured fi m fhe positive x axis
Fmfher,
y=RsmO
wh re R is fhe ~adms of fhe cylinder
(3)
Fcr th present problem, fhe domain of mterest fcr
O'wouldbe :rsO's 2
Th refcre, w d flne c new varicble O as
0=0' :r
so f tt fhe domcm of mterest of O cone pondmgly
bee mes O s O s :r /2 which is more convenient fcr
arurlysis
b hod cmg(2) & (3) mto (I),w obtain
p=pa,,+OSpU2(1 4sin20)+pgRsmO (5
N w n m dbm nsicrurlismg th dist mces with R md
fhe p~essmes with O SpU~ md retami g th same
notatiom for noedimff~sionalised variables, w get
p = Pab~ + (I 4 sin O )+ F ~
d
wh re Fd is d aft based Froude mmmber defmed as
F U~
Eqrction (6) gives fhe prevailmg p~essme
dishibution on th submerged part of fhe cylmder
contour at th mset of fhe g c ity flow The maximc
md mmimc of fhis p~essme d6 tributi m cm be
(~
obtained by fhe cor~ntiorul method ie putti g
P =0
dO
which leds to
cosO ( 3sinO + ~ )= 0
The points of optimum pressure will be given by
0 = 2 md sin0 = ~
(g)
It is import mt to note fnat in fhe second case, fcr O
to be memmgf 1, one mmst have
Fd >0 5
(9)
However, th re is no re triction m Fd bei g large
even upto oo Frocedmg f th r to ch ck the maximc
md mmima, w fmd fhe second derivative,
(4) d~p zo 2smO
~ = 3+16sm
dO Fd
(1 o)
Evaluati g (10) fcr fhe vulues of O obtamed m (3), w
get
P = g _ for 0 =—
dO~ Fd 2
(11)
d P= g+_4 forO=sin ~— (12)
d O 2 Fd 4 Fd
Keepmg fhe resh icti m given by (9) m mmd, w flnd
fnat (11) gives c positive defmite md (12) gives c
negative defmite value for Fd ~ 0 5 b of her words,
fhe pressure has c maximmm at 0 = sin (1/ 4Fd )
md cmmimum ctfhe bottom mo t point 0 =:r/2 m
fhe domain OsO s:r/2 it may be pomted out fnat
for Fd = 0 5, (11) md (12) bodh give zero, co fumi g
(9) in fact for Fd = 0 5, only one maximmm occurs
et O=:r/2 mth domcm OsOS:r,whilefor Fd
more th m O 5, fhere is c cone pondi g maximum et
:r O Hence w conclude th~t for Fd '0 5, th re is
maximum m fhe pressme d6 tributi m et
0 = sm ~ (1/ 4Fd ). md ob iously this poi t is the
OCR for page 545
tbm rged stagnation point Th pressme decreases
on bodh sid s of it, so the velocity mu t be zero
Table I placed in fhe section on Bmw Dmg) gives
fhe locati m of th SSP md fhe cort pondmg vallt
oft ndimertsionalpt s tt cceffcientdefm dinth
conventimurlwayas Cp = P Pat )/0SpU~
different vulues of F~ it may be observed ft~t for F~
=05,th SSPisat O=:r/2,th Imvestpointofth
cylmd r As Pt inct cses, th SSP moves upward
towards th fiee smtcce md flnally for Fd =°°,
0 = 0, i e SSP coincides with fhe t tgnation poi t of
d tble body flow, th n p~,,, is duly t placed by p~
fhe defmition of C~ This t tit will be discussed
f th r in c latter section m inerticl gm itatimur
balance
PresmreProDde cud Ftow witb SSP
— Ow C .] O _ o
~ ilss~
~ \ ~ss~ 0 /
N\, ~to- /
~1 ~
\
Fig 2 Prosscro distriLctioc beforo (t = 0 ) sad st tho ocsot
of gravih tow (t = 0~) for tho dow psst s
hortrochl somisubm rgod circclsr cylindor Tho
mdisl distanco m sscrod from tho body givos tho
v~lue of Cp
A t piccl pt s tre di tributi m befot md et
fhe omet of g c ity flow, for F~ = 0 5 md I O is
show t m Pig 2 The pressut coeffcient at fhe pomts
on th co t tr is givert by fhe mdial distance fi m
fhe pomt to fhe cmve it may be obsetved ft dt m th
case of double body flow (t=O:, th pt ssut
umfommly decreses from A to D, but et fhe mset of
gm ity flmw i e t=0, the pt s tre increses fi m A
to SSP md th n d et ases to D The SSP oc trs et O
=1445°forF~=10 Thept s tt dct csesonbodh
sides of fhe SSP Conseq ently, at fhe onset of
gm ity flow, fhe fluid particles, above th SSP will
mm upwards md below fhe SSP will move
dm~rds Th upward mm mg tt am will t verse
owmg to fhe limitati m d e to glavity, t tltmg m th
PSSP md bow vo tices provided th t face is not
brokert, as expkmedin mearlier secti m
Celenhtdou of FSSP
The PSSP is creted when th flow fi m
SSP to P Pig 1) t verses md its velocity comes m
balance wifh th incomi g flow velocity clo g th
fie trface The velocity of th t verse flow at th
poi t A (fhe stagnation pomt of the d tble body
flow) et th mset of glavity flow m be cclculated
by mplymg Bert ulli's eqtwti m et fhe SSP md et th
poi t A The fommer bemg c stagnati m pomt of th
gm ity flmw, fhe velocity is zero th re md fhet fot
w get
P)i = P)~ +V
(13)
where V is th velocity of th upward flow et A Th
pt ssut P)DSI! md p)~ c m be obt tined by
tb titutmg 0 = 0D~, md 0 = 0 t sp ctively m
eq ction (6) Accord6 gly, w obtam
V2=
4F~
(1 4)
So far w hcve ben obtammg fhe t tits just by
cnalysmg th imticl condition, es it was pe t tim g to
fhe mset of glavity flow m immediately fhereafer;
but th next tep i e bahmci g fhe t verse flow
cgam t fhe mcommg flow clo g fhe fie trface, is c
t suit belongmg to fhe flnal tecdy state, which c m
be achieved mly cider severcl smeller tim steps
fi m th time t = 0+ onward b fhese steps, th fiee
trtcce conditi m is to be cpplied on fhe fiee smtcce,
fhe mcommg md reverse flow is to be cclculated
clo g fhe ft e su face which itself is m urJmow t Of
fhe problem The t n Imearity of fhe ft e smiace
c mditi m tdd c futher complicatiot However,
sit th cim of fhe p mer is to present fhe fhemy md
e pkm fhe m chmism of bow vortices genemtim,
md t t to go into det tiled c mputation, w shell tb
to obttm some cppro imate re tits so as to get m
msight i to fhe phertom nm Accordmgly if ;~ is
fhe slope of th fiee smiace w m w ite th
balancmg process es
COS ;~ = Uz
(1 s)
OCR for page 546
wh re U~ is th x velocity component of th double
body flow slo g fhe x axis
Substitutmg U~=l (I/x2) fmth caseof
fhecacularcylmdr,(l5 leldsto
~ vssr = 1~
As poi ted out, it is not possible to Imow th cmt et
value of ;~, so w shall assmme het ftmt ;~ = 0,
which msy be valid for t latively small values of
Froud mmber Cmseqtt ntlyw obtsin
| 2Fd
d
(17
Fm obtsmmg rel vallt s of x fi m (17 w mmst
have
Fd >— (= 0 71)
V:
(I g)
Hence fhe vulidity of (17 starts from 071 mteld of
O 5 which is fhe natmal re triction for fhe csse of s
cit ular cylmd r qn 9) it msy be noted ftmt th
validity rightly improves ff (16) is used m place of
(17) b this case, w obt tin in placed of (I g)
Fd >
47
(1 9)
Smce cos;~ < I mywh t on th fie t face, th
rhs will slways be less thm 1/4i, md fm
;~ = 60, fhe t sult Fd >1/2 is obtsmed But, as
mentioned earlier, th fiee su face slop md th
velocities of th incomi g md reverse flmws me to be
cal tlated togedher, so myhmg less f m fhe full
flow c mputation will not do md my sppro imste
value of fie smisce slop will lesd to m enoneous
t sult Thet fot, w accept fhe t d ced validity of
fhe fmmuh (17) for fhe FSSF for th pt sent
C omp ens o u wit b E spenmeut sd Results
Th mtmerical values of xp~ ss obtsmed
fi m (17) are d Iy conve ted to d (=x I)Fig 1,
md plotted slo g with fhe experimental vulues in Fig
3 A trve has beert passed f oughth e perimental
poi ts of b'myo et al md is show t tgsm t fhe trve
obt tined fi m present fheory The observatiom c m
be smmmed up ss follows
~ ~~~= ;4
52 OG OS Ot 10
Fig 3
Free smtace sepsmtim poict (FSSF) ic the t ow
psst s semisubme ged horimchl cirmEr
cylinde~Experimect md Theo y
(i) The FSSF moves closer to fhe body m
conflmmstion with th exp rimental re tlts
qttalitatively
(ii) The exp rim ntal vallt s fm Ft < 0 5 se m
to be flt t stmg md do not behave m s
t gular fashion ss fhe values for Ft > 05
This msy be related to th fact ftmt s w 11
defmed SSF does not e i t for Ft < 0 5 ss it
doesforFd>O5(cf Bqn 9)
(iii)
The th ot tical values for Ft = 0 72 to 12
cm be ssid to be ressotmbly close to th
e perim ntal trve keepi g fhe cmde basis
of fheu derivati m m mmd
luer~dld-Gre titedomtl Effect
It is intere tmg to obsetve fhe ch mge m the
location of fhe SSF md the FSSF wifh fhe met sse m
Ft As Ft is mcressed, th SSF mm s closer to the
fie trface md fhe FSSF moves closer to fhe body,
Fig 4 Th region, which is obt tined by jommg fhe
SSF, th FSSF md F (obt tined by taki g bmw wa~
OCR for page 547
height ss 0 5 F42 ) which is s t gi m of tmpp d fluid
contsmmg bow vortices, sh irJcs wifh fhe mct ase of
Ft
Fig 4 Vsrisdon in the bow vortices region with F4 The
region is torm d by ~oining SSF iocstions t the
corre Fonding FSSF iocstions sad hidDg the nse
ot ft e su face = 0 5 F4 tt the cylmder
This c m be expl tit d ss foil ws: Th origm
of fhis happ d vortex t gion is fhe effect of glavity
Th Froude m mber represents th r ttio of inertis
fot e to glavity fot e Vhen Ft is it essed, th
gm ity effect t Ittively red ces md fhet by th
t gion shmks b fhe limiti g csse of Ft = oo th
t gion fmally mishes with fhe SSF commg st 0 = 0
md th FSSF also coming st p = 0, i e m th body
Th fl w thert behaves like double body flow This is
quite logical, ss m fhe sbset of glavity, fhe flow
should behave like double body fl w Th bow wa~
height 0 SF42 becomes mfmite but fhis is dtt to th
fact ft~t Ft is th d tft based Froude m mber md for
Ft ~ °°, w mmst have d aft bec mmg zero, which
mdic ttes ft gt fhet is no gravity actmg my mot md
so no bow wave
BowDrag
Smce fhere is s t verse flow betwen th
poi t A md th SSF Fig 1), fhet will be s 1oss of
pt ssut md sssocitted b w dag, which cm be
obt tined by mtegmti g th x component of th force
dtt to pt ssme fiom A to fhe SSF Accordmgly, th
b w d sg D is given by
ssr
D= | P Pat )n~ds
wh re n~, is th x duection cosine md d is fhe mc
cl me t
Het n~=cosO mdth vallt of P Pat mbe
tb tituted fi m (5 By doing so w obt tin
CD I = | (I 45iD 0+—
2 pU~B O [d
_, sin0)cosR dO
(201
which on perfommmg fhe t quit d mtegmti m md
usi g (8) for 0ssr, gives
d [ d ]
(21)
The sbove formuh c m slso be wwitten in terms of fhe
SSF mdth Froud mmber,ie
C I [ I + g ~ 0 ]
The sbove fomm is, m s way, c mparsble to th one
obt tined by Dsgert md Tulin (I 972, B m 72) One
c m slso expt ss CD as s f nction of fhe SSF only md
obt tin
CD = sin°sss [I + 3 sm 05sS ]
(22)
(23)
Table I gives fhe values of CD for diffet t Ft values
md slso 6he cort pondmg locstion of th SSF The
obset ttions me as foil ws:
e
90.00
44.00
JO.67
23.tt.3
18 00
. 1447
1193
1000
6.37
4.97
o~333
F.
0.5
0.6
0.7
0.8
0.9
10
1.1
1.2
1.5
1.7
20
~ _
Tshle I The iocstion ot subm rged st gnsdon point (9sw)
with the couesponding vslue ot S st the onset ot
grwth t ow sad the nOn dim nsionsl vslue ot
bow d~g (CD) tor di ferent v~ues ot F4 tor the
csse ot semi subm rged hortront~ circulsr
cytinder
C
5 tl0
2.93
204
t 33
1.25
1.17
112
1.05
1.03
102
100
CD 1
3 667 1
1.588
o 864 1
0 552 1
0.387
0.292 1
0.234 1
0.115 1
0 088 1
0~
o.oo 1
I The mtmerical values in Table I sh w 6~t CD
dect sses when th d tft Fr tde mmmber
mct ases, which w tld m m 6~t higher th
d tft Froude m mber, 6he lower 6he b w d tg At
6he fust mstmt this does not sppesl to th
c mm m und rstfmdmg of d sg vs peed
t latiomhip But, it c m be i terpt ted simply m
OCR for page 548
temms of mertial g~avibtiork~l effect as discussed
earlier As F~ inc~eases, th me tie effects
~ektively rise over g ~ lty effects md bow d ag,
bei g ~ comeqrence of gravity effect, ~educes
This is f th r clear fi m fhe location of fhe SSP
which moves closer to fhe fiee smface wifh
mmease of F~ md fhus reduces fhe r mge SSP to
A Fig 4) which is ~e pom ible for inc~easi g th
kmetic energy of th surroundmg water by
gff~eratmg ~everse flow md subseqre tly bow
vortices/bow jeVwhite water
2 Th re is no ekborate experlme tal data a~ilable
to c mpare fhe values m Table I However, Baba
(1969) hes suggested ~ two dbm nsiom~l
~ep~esenbtion of heakmg wa~s for his
e perlme ts on ~ t mker, as ff it was mmiform md
normal to th bow, md has estimated its
eqflvale t le gfh as roughly helf fhe beam
Dagff~ & Tulin 1972) Th d ag coefficie t, per
umt lengfh, cone pondmg to ~ two dbm nsiom~l
fl w across th b~eaki g wave for F~= 1 7 is
given as CD (= Dl O SU~ T)= 0 08, wh re D
is th d ag md T fhe d ~'d Baba 1969 § 7 3)
b teresti gly, w fmd fnat our value of CD for F~
= 1 7 comes out as 0 088 Table 1), which is
very close to fhe estimate of Baba This
closeness of CD vulue obtained by fhe p~esent
fheory with th one estimated fiom e perim nbl
~esults f th r pro id s support to fhe SSF th °b
e pkmmg fhe mechmism of b w votices
gff~eration it may be ~elevmt to m ntion fnat
Dagen & Tulin (1972) obtained fhis value of CD
= 0 17 (about two times) for ~ ve tical step by
sol i g fhe gravity fl w usi g fhe method of two
pert rbation exp msiom
3 Bow d ag dep nd upon fhe location of fhe SSF,
fhe low r fhe SSF th mme is th bow d ag Th
location of fhe SSF on th oth r h md is, d6'ectly
~ekted to th p~essme ddshibutim of double
body fl w on fhe bow, which m t rn dep nd
upon its shme The~efme, th double body
p~essme d6 tributi m on th b w md fhe locati m
of fhe SSF pro id fhe key to fhe desig of
b w co tour for minlmmm b w d ag
Bow Cout~mr Deslgu -two dlmeusloual ease
A bow contour for th two dlmemiorul case
m be designed now for mmlmum b w d ag as
foll ws Th total pressure at my pomt of th bow is
fhe sum of double body pressure md fhe gm lty
p~essme, i e
Cp(~ly =cp(d6) +cre
The c mdition for fmdmg SSF is given by
aCp(b~l) O
ay
which gives
aCp(d&) =
dy
(25)
acpe
dy
b gene~al Cp(d &) is maxlmmm at fhe double body
tagnation pomt DBSF) md decreses toward th
bottom as fhe fluid accelerates The g ~ lty pressure,
on th of her hmd is zero at fhe mmdist rbed fie
surface (i e at DBSF) md mcreses toward th
bottom Owl g to fhe opposite sigm of fhe rate of
chmge of fhese two p~essmes, ~ maximum m
Cp (~1) (i e fhe SSF) occms at fhe poi t whe~e th
two rates of chmge me eqral in magmt de, makl g
fhemteofchmgeof CP(~I) tobezero
(26)
The rate of ch mge of double body p~essure
is indep nd nt of F~ md depends pmely up m th
geometb or fhe slopes of fhe bow co tour at d6ffe~ent
poi ts On th of her hmd fhe mte of chmge of
gm lty p~essme is 2/Fd which is comtmt
dependmg upon daid based Froud mmber Th
SSF location is given by fhe mtersection of fhe two
sides of (26) when plotted wifh ~esp et to y Now by
usi g ~ double body flow calcubtion prog amme by
suibble panel medhod md coupli g it with ~ curve
design progmmme usi g Bezier curve B splme wifh
~ visual interface, it should be possible to d sig
b w contom which gives fhe SSF locati m as near th
fi e smface as possible for ~ given value of F~ it may
be desuable to tag th process wifh the selection of
design d fd of th ship This will give fhe scope of
m mipubtion of fhe mte of ch mge of g ~ lty p~essme
additionally Thus th locati m of SSF acts as ~
conhol hmdDe for fhe desig of m efhcient bow
contour md also serves as ~ measme of pe formance
with regard to b w d ag
Foracacularcylmdr Fig l),w cmwite(25)as
CP (~U) = 1 4Y ~ Y
md mplymg fhe procedme as descril~d above based
on (26) fhe location of SSF is shalghbway obbmed
analytically as
(27)
OCR for page 549
Fig 6 shows th location of SSP m fhe case
(2g) of th ve ticcl step for differe t d cid Froud
m mbers
4 F~2
l
Y=
Th cpplicati m of fhe above to of her shapes is shmvrt
belmw
SSP for Ve ttieel Step cud Bulbous bow
So far w have ben usmg c t gmbr shap,
ie c cu thr cylinder, for fhe cpplication of th
proposed fheob it is so becmse w hcve th
experime tal results fm th cu tlar cylmder But th
fheory c m be cpplied to of her bmw shapes tually
w 11 md whatever t suits are obt tined arulyticclly m
fhe case of th cu tlar cylmder, cm be obtamed
m mericclly in fhe case of t m gular two
dimertsional bow shapes As m example, w pt sent
here fhe application of th th °b to c ve tical tep
md c bulbous bmw shcpe Th geomeby of th
verticcl step md th bulbous bow hcs beert ttkert
fi m Grosenbaugh md Y tog(l959) The d tble
body flmw is cclculated usmg c low ord r panel
method Fig 5 show th cmves of C (tb) md th
C (totcl) for Ft=l
Fig 5 The vsnstioc of C~(db) sad C~(~t) vs Dmd
forvert(cs( s step sad Bc(bocs bow
Th shmpchmgesinth slop mditsduectimmfhe
geometb are d Iy reflected m fhe culve of C (tb ) at
fhe cmt spond6 g pomts The maximc occunmg m
fhe culve of totcl pressut m be esily marked
s,
Wi ~
t, t
t, j
F(g 6 Determ(mt(oc of subm rged shgnst(oc po(cts for s
vert(cs( step The vert(cs( ((ces show the s(ope of
gm ih comFocect of presscre
Det rmicst(oc of submerged shgnst(oc po(cts for
s bc(bocs bow The Vert(m( ((ces show the s(ope
of grw(h comFocect of presscre
determit d by fhe procedme based on (26) md
described in th earlier section i e by d cwmg th
slop cmve of C (tb ) md fhe con t mt slope Imes of
C fm d6fferent vallt s of Ft The i tersecti m of the
two slope ttves fhert gives th cmrespond6ng SSF
locatiot it c m be seen thct es Ft mcreases, fhe SSF
m th ve tical tep moves closer to fhe fie smtcce,
simibr to fhe case of c cit ular cylmd r For Ft = 1,
fhe SSF lies c little above half th d cid, which is
c mparable to th value es show t by Grosenb mgh
md Y t g (I 9S9) with fhe flmw computatiot Fig 7
shows fhe sarne for fhe case of bulbous bmw shcpe
Owmg to th fi tuent chmges in fhe dbt ction of
slop m fhe shap, fhe maximc me not es w 11
OCR for page 550
defned es m th case of fhe verticcl step For
example, for F~ = 1, fhere are two maxlmc md two
mmlm~ Accordmgly th flow pict ~e will be mme
complicated f m as hcs l~en d scribed for the case
of fhe cucular cylmd r m verticcl step But, smce th
see md maxlmc hcs fhe higher value of C Oobr) fnat
should be bken es th hue maximum md be fhe SSF
m th cunent context it is fhen observed fnat th
SSF for fhe bulbous bow shcpe is low r thm m th
case of fhe verticcl tep for fhe same value of
F~ = I This provides m exphnati m for th
observation ~epmted by Grosenbmgh md Yeu g
(1959) fnat th wa~ hekmg is delcyed for c
bulbous b w relative to c ve tical step, smce low r is
fhe SSF fhe less doml mt is th mertic effect md
wa~ brekmg which is duectly Imked wlfh sped is
debyed accordmgly
Bow Vor~des - tb ree dlmeuslousd eese
Th foami g motion or white water
observed at fhe b w of c ship md cll clong th load
waterline k wn es neckkce vo t x, is th thee
dlmff~sional pict ~e of bow vortices if th bow
vmtices me genemted in fhe two dim nsicrurl case by
fhe presff~ce of m SSF l~low fhe fiee su face, it will
be just right to conject ~e fnat m SSF may xist et
fhe ship's b w md et th secti ms l~low th fie
surtcce b of her wmd, th curve of C omr) vs d aft
may have c maximum et th b w md et fhe sections
Smce fhe ship has c fmite d cid, its double
body flow must acqune c c mpone t of velocity m
fhe depfh duection right fi m fhe forward
perp ndlcukr down th hull Accordmgly, et th
b w fhere is c d mecse of double body pressure fi m
forward pemff~dicukr to th kel md et each secti m
fi m load waterlme to th bilge, which in presence of
gm lty, ~esults m c SSF bel w th fie smtcce
givmg rise to c fl w fi m fhe SSF to fhe fie smtcce
This upward flow may fmm c jet et the fiee smiace,
enbam c lot of air md dismteg ate i to immm rable
bubbles at fhe fiee su face ~eflecting mme light
owing to their k ge smtcce are md form c white
water or c foam lik mpe ance
Coueluslous
Th problem of b w vo tices genemtion et
kboratmy scale is cdd essed h re with fhe mcm clm
to e pkm fhe ~esults of fhe experiments on c
hml ontal semisubm rgedcacular cylmd r, md fhe
mechmism of bow vortices genemti m observed
ahecd of fhe cylinder
I it is shown fnat fhere xists c tagnation pomt
bel w fhe fie suface, ie c submerged
sbgnation poi t (SSF) which is resp msible for
makmg c bmnch of fhe mcm t~eam flow
upward towards th fiee su face md fhus
produce c ~everse fl w which results m bow
vmtices md c f~ee surface sepamti m pomt
FSSF)
2 The SSF is c f nction of d cid based Froud
mmber F~ md double body p~essme
dishibutioa The SSF occms whff~ th double
body pressure d meases along th d cid md
c msequently, c maxlmmm mpears m th tobl
p~essme below fhe fiee smtcce it is fmfher
discussed fnat th location of the SSF represents
c balance l~tween mertial effects md g~avity
effects The SSF moves towmds fhe fie
smtcce as F~ is mmeased md only fm F~ = oo
(i e no g c lty, mly inertlc) SSF lies on fhe fiee
smtcce comcidmg wifh fhe sbgnati m pomt of
double body flow
3 The values of FSSF cclculated on th basis of
fhis fheory for flow past c horl o tal semi
submerged cacuk cylmder e pkm the
e perim nbl ~esults of b'myo et al (1952)
qualibtively md q mtitatively to c recsonable
e tent
4 The bow d cg is obbmed es c fm tion of d cft
based Froude mmber The value of the bow
d cg for F~ = 1 7 for fhe case of c cimuk
cylmder, cgres very w ii wlfh th value
e tlmated by Bcbc (1969) by e perim nts on c
tsrD~r md fmdmg c two dim nsiorul equivalent
of fhe same
5 It hcs beff~ conject red thct fhe bow vo tices m
fhe f e dlmemiorul case, ie th neckkce
vmtex m c ship, me also produced by fhe sarne
c msideration Owmg to c fmite d cid, fhere is c
c mpone t of velocity m fhe depth duection m
double body flow of th ship which lecds to c
decrecse of double body flow p~essme fi m
forward pemendicuk to keel et fhe bow md
fi om waterlme to bilge at sections This results
m c SSF et fhe b w md et fhe sections below
fhe f~ee smtcce Th flow from fhis SSF
towards fhe fiee su face results in c jet et the
fiee su face, which ent~ams air md fmms
bubbles makmg white water m c foam like
cppeamnce
OCR for page 551
Scope of tbe future work
I C mputstiorul st dbes are ~equned to be
undertakff~ to verify fhe ~esults of this th °b
Fu t, fhe fl w past c horizontcl se iisubmerged
cucukr cylmd r should be computed md th
SSP location, ~eversal of flow, th FSSP
location md fhe bow d cg be calculated md
checked wifh fhe results obtcined by th fhecry
2 The experime ts m c horizontcl
semisubmerged cacukr cylmder should be
~epeted md fhe bow vortices should be
visualised md st dbed m fhe light of the p~esent
fheory The SSP location should be found md
fhe m~t ~e of reverse fl w should be st dbed
Th PSSP locati m should be found md
compared with th fhecry The oscillatiom m
fhe PSSP locati m md cone pondi g bow wa~
height should be st dbed Also, fhe b w d cg
should be found exp rim ntclly md checked
withth fheory
3 Th steps (1) md (2) should be c mducted fcr
odher bow ge mehies with different slop
dishibution, e g verticcl step, conventicrur
ships bow md bulbous b w et
4 Th conjectme mcde fcr bow vo tices
gff~erction in c f ee dimemiorul case is to be
verifled by usmg c the dbm nsicrurl arulyticcl
body generated by c k own combination of
singmbrities The flow past c sphe~e or Rmkine
ovul do not serve fhe pu pose es th se me
axisymm hic Th c mbination should be such
as to result m c hue f ee dimff~sional fl w
This will ~equne fhe dishibution to be
asymm hic Alternatively, fhe double body
p~essmes may be computed m c Wigley hull
usi g c higher ord r pmel method so fnat th
poi ts et fhe bow c m be tsken as nodes es w 11
as collocation pomts md fhen th p~essme
dishibution et fhe b w md at th section should
be obbmed Subs quently, fhe SSP et fhe bow
md et fhe sections should be calcubted to
conflmm fhe genemti m of bow vertices in th
f ee dimff~sional case We have worked wifh
fhe values of C for double body flow for c
series 60 h 11 obtcined fiom "SB P PLOW" but
fhese w ~e et th panel cc troids which w ~e
cway from fhe pomts at fhe b w contour
Owmg to fhe high tmgential velocity m th
neighbourhood of bow, fhe C was far bel w th
expected value of umty et DBSP We used c
four/five deg~ee smiace flttmg but th
exhapokti m was not sati factory
Refereues
Bcbc, E "A New C mpone t of Viscous Resistmce
of Ships" Journal of the Socieh of Ncval A chitects
of Jcp m, Vol. 125,1969, pp 23 32
E kc t, E md Sharmc, 5 D "Bugw I te fur
Lmgsame, Vollige Schiffe" Jahrbuch der
Schiffl mtechmisch n Gesselschcf Vol. 64, 1970,
pp 129 171
Smuki, K, "On fhe Drag of Two Dim nsicrurl
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Representative terms from entire chapter:
fhe ssf
